ProLog

In our last post on Expected Values, we covered some of the basic tools a pollster might use to analyze a single random variable.  Before we move on to comparing 2 RV’s, we first need to cover one more preliminary concept.

Probability

Probability is a concept that’s easy to understand, but difficult to master.  It is a number between 0 and 1, assigned to an event, that indicates the likelihood of that event occurring.  An event is a collection of outcomes from a given RV.  The event of rolling an even number on one throw of a fair die includes the outcomes 2, 4, and 6.  An event with 0 probability means there’s no chance it will occur; an event with a probability of 1 means that it will occur with absolute certainty.

We calculate probabilities of events when we don’t have a procedure to predict the outcome with certainty (as with the trajectory of a rocket, for example).   Calculating them is a mix of art and science; a variety of mathematical rules are available to help you, but in the end, it’s a matter of analyzing all the available information.

One such  rule is, if all outcomes are equally likely (i.e. a uniform distribution), the probably of an event is the number of outcomes that satisfies the event, divided by the total number of outcomes.  The probability of rolling a 5 on a die is 1/6 (0.167) since only 1 outcome yields this result and there are 6 possible outcomes.  The probability of throwing an even number is 3/6 (1/2 or 0.5) since there are 3 outcomes that yield this result.

Probability Functions

We denote the probability of event A as Pr(A).  We can also define a probability density function (PDF) for continuous RV’s, and a probability mass function (PMF) for discrete RV’s.  PDF’s and PMF’s plot the probability of each outcome in a single function.  Many common functions occur frequently in nature.  One of the most common is the Gaussian or Normal function show below.

courtesy of wikipedia.org

The Normal PDF forms the well known “bell-shaped curve”, with it’s mean value, µ, at the center.  We’ll return to the Normal function later, but for now will confine ourselves to discrete PMF’s.  For our die example, the PMF is:

coutesy wikipedia.org

Here we see that the probability of each outcome is the same, 1/6.